Dimensional dual hyperovals associated with Steiner systems

In Adv. Geom. 3 (2003) 245, a class of d-dimensional dual hyperovals is constructed starting from a subset X of PG(d,2) with certain properties. In this paper, a criterion for X to provide a d-dimensional dual hyperoval is given in terms of some functions. Based on this, we describe such subsets, and show that there are exactly two isomorphism classes of d-dimensional dual hyperovals arising from those subsets and that a similar statement holds for those of the associated Steiner systems S(3,4,2d+1).